THE THEORIES OF FUNCTORS AND MODELS

Abstract

This chapter discusses the theories of functors and models. Domain is the category of topological-spaces-with-base-points-and-continuous-base-point-preserving-maps, and its range is the category of groups-and-group-homomorphisms. It is precisely these properties that are formalized into the definition of functor. But first we must have a definition of category. The shortest known definition is so short that it is misleading. The objects of a category are usually thought of as sets with some sort of structure and the maps as functions which in some way preserve the structure. But note that the definition of category is a first-order definition. In most categories in nature the difference kernel of two maps is constructed as the set of elements in their domain on which the two maps agree. The dual notion, that of difference cokernel does not have a standard construction. The fact that these definitions do not always agree with the usual notions has turned out to be not a disadvantage but one of the main attractions of category theory.